Nonlinear dynamics of a ball rolling on a surface


An underlying potential energy function can provide visual and intuitive insight into a system's stability and overall behavior. In particular, the motion of a ball moving along a curve or surface in a gravitational field provides a macroscale demonstration of interesting dynamics. We investigate the motion of a small ball rolling along a smooth two-dimensional potential surface. A direct experimental realization of this situation is suitable for demonstrating some classic features of nonlinear dynamics. The results of numerical simulations are directly compared with experimental data. To better characterize the dynamical behavior of the ball, especially when it is undergoing chaotic motion, several descriptive measures are discussed, including time-lag embedding, initial condition maps, power spectra, Lyapunov exponents, and fractal dimensions. © 2010 American Association of Physics Teachers.